Sakai’s theorem for $\mathbb{Q}$-divisors on surfaces and applications

Abstract

In this paper, we present a characterization of a big Q-divisor D on a smooth projective surface S with D2 > 0 and H1(OS(−D)) = 0, which generalizes a result of Sakai [Sak90] for D integral. As applications of this result for Q-divisors, we prove results on base-pointfreeness and very-ampleness of the adjoint linear system |KS + D|. These results can be viewed as refinements of previous results on smooth surfaces of Ein-Lazarsfeld [EL93] and Ma¸sek [Ma¸s99].